# Properties

 Label 2-48e2-8.3-c2-0-45 Degree $2$ Conductor $2304$ Sign $0.707 + 0.707i$ Analytic cond. $62.7794$ Root an. cond. $7.92334$ Motivic weight $2$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 2i·5-s − 6.92i·7-s + 6.92·11-s + 2i·13-s − 10·17-s + 20.7·19-s + 27.7i·23-s + 21·25-s + 26i·29-s − 6.92i·31-s − 13.8·35-s − 26i·37-s + 58·41-s + 48.4·43-s + 69.2i·47-s + ⋯
 L(s)  = 1 − 0.400i·5-s − 0.989i·7-s + 0.629·11-s + 0.153i·13-s − 0.588·17-s + 1.09·19-s + 1.20i·23-s + 0.839·25-s + 0.896i·29-s − 0.223i·31-s − 0.395·35-s − 0.702i·37-s + 1.41·41-s + 1.12·43-s + 1.47i·47-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$2304$$    =    $$2^{8} \cdot 3^{2}$$ Sign: $0.707 + 0.707i$ Analytic conductor: $$62.7794$$ Root analytic conductor: $$7.92334$$ Motivic weight: $$2$$ Rational: no Arithmetic: yes Character: $\chi_{2304} (127, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 2304,\ (\ :1),\ 0.707 + 0.707i)$$

## Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$2.223576312$$ $$L(\frac12)$$ $$\approx$$ $$2.223576312$$ $$L(2)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
3 $$1$$
good5 $$1 + 2iT - 25T^{2}$$
7 $$1 + 6.92iT - 49T^{2}$$
11 $$1 - 6.92T + 121T^{2}$$
13 $$1 - 2iT - 169T^{2}$$
17 $$1 + 10T + 289T^{2}$$
19 $$1 - 20.7T + 361T^{2}$$
23 $$1 - 27.7iT - 529T^{2}$$
29 $$1 - 26iT - 841T^{2}$$
31 $$1 + 6.92iT - 961T^{2}$$
37 $$1 + 26iT - 1.36e3T^{2}$$
41 $$1 - 58T + 1.68e3T^{2}$$
43 $$1 - 48.4T + 1.84e3T^{2}$$
47 $$1 - 69.2iT - 2.20e3T^{2}$$
53 $$1 + 74iT - 2.80e3T^{2}$$
59 $$1 + 90.0T + 3.48e3T^{2}$$
61 $$1 - 26iT - 3.72e3T^{2}$$
67 $$1 - 6.92T + 4.48e3T^{2}$$
71 $$1 - 5.04e3T^{2}$$
73 $$1 - 46T + 5.32e3T^{2}$$
79 $$1 + 117. iT - 6.24e3T^{2}$$
83 $$1 + 48.4T + 6.88e3T^{2}$$
89 $$1 - 82T + 7.92e3T^{2}$$
97 $$1 - 2T + 9.40e3T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$